MIRR for Nonconventional Cash Flows

Why Use MIRR for Nonconventional Cash Flows?

A conventional project often begins with an outflow and then has only inflows. Other projects alternate between outflows and inflows: a project may require a later cleanup cost, a major equipment replacement, or a further investment after initial revenue. These are nonconventional cash flows. Their changing signs can produce more than one IRR, so an IRR alone may not provide a unique return summary.

The modified internal rate of return (MIRR) handles this problem by using one specified rate to bring negative cash flows to the beginning of the project and another to carry positive cash flows to its end. It then reports the single annual rate that connects those two equivalent amounts. MIRR is a summary measure, not a replacement for examining the project’s cash flows and value.

Definition: Let $C_t$ be the project cash flow at the end of period $t$, for $t=0,1,\ldots,n$. A cash flow is nonconventional when its signs change more than once over the project horizon. Given an effective annual finance rate $f$ and an effective annual reinvestment rate $r$, both greater than $-1$, define $$ P^-=\sum_{C_t<0}\frac{-C_t}{(1+f)^t}, \qquad F^+=\sum_{C_t>0}C_t(1+r)^{n-t}. $$ When there is at least one negative and one positive cash flow, the MIRR is $$ \operatorname{MIRR}=\left(\frac{F^+}{P^-}\right)^{1/n}-1. $$ Here $P^-$ is the time-0 present value of negative cash flows, and $F^+$ is the time-$n$ future value of positive cash flows.

The rates are assumptions supplied for the analysis; they need not be equal. The finance rate represents the rate used to value required project funding, while the reinvestment rate represents the assumed return earned on positive cash flows until the project horizon. Cash flows are assumed to occur at the stated period ends, with rates and periods measured consistently. If a cash flow stream has no positive cash flow or no negative cash flow, this MIRR formula does not define a project return.

Why the Formula Produces One Rate

Once the negative cash flows have been discounted to time 0 and the positive cash flows accumulated to time $n$, the original stream has been summarized by one outflow $P^-$ and one inflow $F^+$. The MIRR is the annual rate that makes those amounts equivalent across $n$ periods.

Value the negative cash flows at time 0.
Discount each outflow at the finance rate $f$, then add the positive magnitudes to obtain $P^-$.

Value the positive cash flows at time $n$.
Accumulate each inflow at the reinvestment rate $r$ to the project’s final period, then add them to obtain $F^+$.

Solve for the annual rate.
Set the time-0 amount accumulated at the MIRR equal to the time-$n$ amount, then solve for the rate.

The justification is the equivalence equation

P^-(1+\operatorname{MIRR})^n=F^+.

Dividing by $P^-$ and taking the positive $n$th root gives the definition. Because $P^->0$ and $F^+>0$, the equation has one solution for $1+\operatorname{MIRR}>0$. In contrast, an IRR is a root of the original cash-flow equation, whose multiple sign changes can produce multiple roots. MIRR does not find or select among those IRR roots; it summarizes the project using the specified finance and reinvestment assumptions.

Worked Example: Applying Different Rates to Outflows and Inflows

A three-year project has cash flows of $-\$600$ at time 0, $+\$500$ at the end of year 1, $-\$441$ at the end of year 2, and $+\$726$ at the end of year 3. The finance rate is 5% effective annually, and the reinvestment rate is 10% effective annually. Calculate the MIRR.

Method. Discount both negative cash flows to time 0 at 5%, and accumulate both positive cash flows to the end of year 3 at 10%. The signs alternate, so treating the cash flows as a conventional stream and seeking a single IRR could be misleading.

The present value of negative cash flows is

P^-=600+\frac{441}{(1.05)^2} =600+\frac{441}{1.1025} =600+400 =\$1{,}000.

The future value of positive cash flows at the end of year 3 is

F^+=500(1.10)^{3-1}+726 =500(1.21)+726 =605+726 =\$1{,}331.

There are $n=3$ periods, so

\operatorname{MIRR} =\left(\frac{1{,}331}{1{,}000}\right)^{1/3}-1 =(1.331)^{1/3}-1 =1.10-1 =0.10.

The MIRR is 10.00% effective annually. Check: accumulating the time-0 equivalent at 10% for three years gives $\$1{,}000(1.10)^3=\$1{,}331$, equal to the calculated terminal value of the positive cash flows. The common mistake avoided here is using the reinvestment rate to discount the year-2 outflow; the stated finance rate applies to negative cash flows.

A Stream with Multiple IRRs

A nonconventional stream can have several IRRs because its present value equation may cross zero more than once. MIRR instead gives each cash-flow sign a specified valuation treatment. The following example shows both calculations so the distinction is clear.

Worked Example: Replacing Multiple IRR Roots with One MIRR

A two-year project has cash flows $-\$100$ at time 0, $+\$230$ at the end of year 1, and $-\$132$ at the end of year 2. Use a 5% effective annual finance rate and an 8% effective annual reinvestment rate to calculate its MIRR. Also verify that the project has two IRRs.

Method. First form the IRR equation in the annual rate $R$. Then calculate MIRR separately using the specified rates; do not treat either IRR root as the MIRR.

The IRR equation is

-100+\frac{230}{1+R}-\frac{132}{(1+R)^2}=0.

Multiplying by $(1+R)^2$ and simplifying gives

-100(1+R)^2+230(1+R)-132=0, \qquad 100R^2-30R+2=0.

Factoring the quadratic gives $(10R-1)(10R-2)=0$, so the two IRRs are $R=10\%$ and $R=20\%$.

For MIRR, the time-0 present value of negative cash flows is

P^-=100+\frac{132}{(1.05)^2} =100+\frac{132}{1.1025} =100+119.7279 =\$219.7279.

The year-1 positive cash flow is accumulated one year to the end of year 2:

F^+=230(1.08)^{2-1}=\$248.40.

Thus

\operatorname{MIRR} =\left(\frac{248.40}{219.7279}\right)^{1/2}-1 \approx 0.06324.

The MIRR is 6.32% effective annually, rounded to two decimal places. Check: $(1.06324)^2$ is approximately $1.1305$, the ratio of the terminal positive value to the present value of negative cash flows. The common mistake avoided is choosing one of the two IRRs and calling it the project’s unique return; MIRR is a different measure based on the stated 5% and 8% assumptions.

Interpreting the Assumptions

Finance and reinvestment rates answer different questions. The finance rate determines how costly later outflows are when expressed at time 0. The reinvestment rate determines how much earlier inflows contribute to the final project value. Using separate rates can be more transparent than assuming that every interim positive cash flow can earn the project’s own IRR.

The timing exponent matters. A positive cash flow at the end of year 1 in a three-year project is accumulated for two periods, not three. A negative cash flow at the end of year 2 is discounted for two periods. All exponents are measured from the time of the cash flow to the valuation date, rather than from the start of the project without regard to the cash-flow date.

Worked Example: Checking Timing and Rate Assignment

A project lasts three years and has cash flows $-\$500$ at time 0, $+\$200$ at the end of year 1, $-\$121$ at the end of year 2, and $+\$478.255$ at the end of year 3. The finance rate is 10% effective annually, and the reinvestment rate is 4% effective annually. Calculate the MIRR.

Method. Discount the year-2 outflow two periods at the finance rate. Accumulate the year-1 inflow two periods at the reinvestment rate; the year-3 inflow is already at the terminal date.

P^-=500+\frac{121}{(1.10)^2} =500+\frac{121}{1.21} =500+100 =\$600.

F^+=200(1.04)^2+478.255 =200(1.0816)+478.255 =216.32+478.255 =\$694.575.

With $n=3$,

\operatorname{MIRR} =\left(\frac{694.575}{600}\right)^{1/3}-1 =(1.157625)^{1/3}-1 =1.05-1 =0.05.

The MIRR is 5.00% effective annually. Check: $\$600(1.05)^3=\$694.575$, exactly the terminal value calculated for positive cash flows. A common timing error is to compound the year-1 receipt for three periods; it has only two periods to grow before the end of year 3.

What MIRR Does and Does Not Tell You

MIRR gives a single annualized rate under explicitly stated assumptions. It avoids the ambiguity of multiple IRR roots, but the result depends on the finance and reinvestment rates chosen. Changing either rate can change the MIRR. A rate is therefore not a property of a cash-flow list alone; it is a result conditional on the model and its inputs.

MIRR also does not, by itself, establish whether a project creates value. A positive MIRR is not a complete decision rule without considering the appropriate required return and the project’s scale and timing. Keep it distinct from NPV, which measures value in dollars using a specified discount rate. In particular, do not interpret the unique MIRR as proof that the original cash-flow equation has only one IRR.

Key takeaway: For nonconventional cash flows, use the finance rate to discount every negative cash flow to time 0, the reinvestment rate to accumulate every positive cash flow to time $n$, and then solve $\operatorname{MIRR}=(F^+/P^-)^{1/n}-1$. State both rates and the project horizon whenever reporting the result.

Check Your Understanding

For each problem, use effective annual rates and report the MIRR as an annual percentage.

A two-year project has cash flows $-\$400$ at time 0, $+\$550$ at the end of year 1, and $-\$100$ at the end of year 2. The finance and reinvestment rates are both 0%. Calculate its MIRR. (Answer: 4.88%)
A three-year project has cash flows $-\$800$, $+\$300$, $-\$242$, and $+\$826.875$ at times 0, 1, 2, and 3, respectively. The finance rate is 10% and the reinvestment rate is 5%. Calculate its MIRR. (Answer: 5.00%)
A two-year project has cash flows $-\$700$ at time 0, $+\$900$ at the end of year 1, and $-\$110.25$ at the end of year 2. The finance rate is 5% and the reinvestment rate is 10%. Calculate its MIRR. (Answer: 11.24%)
A three-year project has cash flows $-\$1{,}200$, $+\$500$, $-\$300$, and $+\$900$ at times 0, 1, 2, and 3, respectively. Both the finance and reinvestment rates are 0%. Calculate its MIRR and state whether it is positive or negative. (Answer: $-2.27\%$, negative)

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